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 考慮一個線性模型，有2 個控制變數及m 個反應變數，其中反應變數之間是相關的，而且共變異數矩陣是已知的。基於上述的模型，有兩個問題是我們感到有興趣的。第一個問題是從實驗中觀察到一組m 個反應量後，想要反推出2 個控制變數所設定的水準為何。對於這個問題我們使用古典估計量去估計2 個控制變數所設定的水準，而古典估計量的數學符號為x_c。第二個問題是對於設定一組想要達到的水準，如何找到一個適合的估計量用來估計2 個控制變數的水準。對於這個問題，我們定義一組最適控制數，這組最適控制數會使得標準化後的設定水準與期望反應的差距平方加權總和達到最小，而這組最適控制數表示成x_T。這篇文章的目的就是對於這兩種估計量分別找到個別的c-最適設計，能使得個別的均方誤差達到最小。接著我們會比較這兩個最適設計差別，並且觀察這兩者分別跟均勻設計相比的效率。
 Consider a linear regression model with a two-dimensional control vector (x_1, x_2) and an m-dimensional response vector y = (y_1, . . . , y_m). The components of y are correlated with a known covariance matrix. Based on the assumed regression model, there are two problems of interest. The first one is to estimate unknown control vector x_c corresponding to an observed y, where xc will be estimated by the classical estimator. The second one is to obtain a suitable estimation of the control vector x_T corresponding to a given target T = (T_1, . . . , T_m) on the expected responses. Consideration in this work includes the deviation of the expected response E(y_i) from its corresponding target value T_i for each component and defines the optimal control vector x, say x_T , to be the one which minimizes the weighted sum of squares of standardized deviations within the range of x. The objective of this study is to find c-optimal designs for estimating x_c and x_T , which minimize the mean squared error of the estimator of xc and x_T respectively. The comparison of the difference between the optimal calibration design and the optimal design for estimating x_T is provided. The efficiencies of the optimal calibration design relative to the uniform design are also presented, and so are the efficiencies of the optimal design for given target vector relative to the uniform design.
 1 Introduction 12 Optimal design for calibrations 3 2.1 The classical estimator xc 3 2.2 The coefficient vector c_β 4 2.3 The optimal calibration design 5 2.3.1 The equivalence theorem 5 2.3.2 The weight of support points 73 Optimal design for given target vector 7 3.1 The optimal control vector x_T 7 3.2 The coefficient vector c_{β,T} 8 3.3 The optimal design for given target value problem 94 An example 9 4.1 The first problem: y is observed 10 4.2 The second problem: T is a given target vector 12 4.3 The comparison of the results for the two problems 145 Discussion 15References 20
 Berkson, J. (1969). Estimation of a linear function for a calibration line. Technometrics, 11, 649-660.Brown, P. J. (1982). Multivariate calibration. J. R. Statist. Soc. B, 44, 287-432.Brown, P. J. (1993). Measurement, regression, and calibration. Clarendon Press, Oxford.Buonaccorsi, J. P. (1986). Design considerations for calibration. Technometrics, 28, 149-155.Cook, R. D. and Nachtsheim, C. J. (1982). Model robust, linear-optimal designs. Technometrics, 24, 49-54.Kitsos, C. P. (2002). The simple linear calibration problem as an optimal experimental design. Communications in Statistics - Theory and Methods, 31,1167-1177Krutchkoff, R. G. (1967). Classical and inverse regression methods of calibration, Technometrics, 9, 425-439.Krutchkoff, R. G. (1969). Classical and inverse regression methods of calibration in extrapolation. Technometrics, 11, 605-608.Lin, C. S. and Huang, M.-N. L. (2006). Optimal designs for calibrations in multiresponse-univariate regression models. Manuscript.Nishii, R. and Krishnaiah, P. R. (1988). On the moments of classical estimates of explanatory variables under a multivariate calibration model. Sankhya Ser. A, 50, 137-148.Oman, S. D. and Srivastava, M. S. (1996). Exact mean squared error comparisons of the inverse and classical estimators in multi-univariate linear calibration. The Scandinavian Journal of Statistics, 23, 473-488.Osborne, C. (1991). Statistical calicration: a review. Int. Statist. Rev., 59, 309-336.Pukelsheim, F. (1993). Optimal design of experiments, Wiley, New York.Shukla, G. K. (1972). On the problem of calibration. Technometrics, 14, 547-553.Sundberg, R. (1985). When is the inverse regression estimator MSE-superior to the standard regression estimator in multivariate controlled calibration situations? Statistics & Probability Letters 3, 75-79.
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